I had a bun (Bam) who chopped his teeth so loud it could be heard into another room. This hapoened when he was super-relaxed and comfo and melted into my mum’s lap. (He loved my mum and always wanted to hop into her lap for cuddles). So the loudness doesnt necessarily mean it’s not purring. Teeth chopping in a poorly bun tends to be accompanied by rather obvious signs of discomfort. Refusal to take a treat, shifting positions often bc cant find a comfortable position or laying/sitting very still in a sort of tense manner. Look up the Rabbit Grimace Scale for facial/head signs that a bun is in distress!
I had one bun (Vilde) who loved sunbathing. He sought out sunspots to lie in. He was an elderly, skinny little bun, very fluffy but without undercoat fur. He cuddled up next to the radiator in the winter.
It’s very, very good that you check your bun for hyperthermia, because it is indeed a risk during hot days. You can place sth cold in the room, like a chilled bottle of water and see if he seeks it out to lie next to it.